Gravitational domain walls and the dynamics of G
GGravitational domain walls and the dynamics of G Claudio Bunster ∗ and Andr´es Gomberoff
2, 1, † Centro de Estudios Cient´ıficos (CECs), Casilla 1469, Valdivia, Chile Facultad de Ingenier´ıa y Ciencias, Universidad Adolfo Ib´a˜nez,Avda. Diagonal las Torres 2640, Pe˜nalol´en, Santiago, Chile (Dated: April 12, 2017)From the point of view of elementary particle physics the gravitational constant G is extraordi-narily small. This has led to ask whether it could have decayed to its present value from an initialone commensurate with microscopical units. A mechanism that leads to such a decay is proposedherein. It is based on assuming that G may take different values within regions of the universe sep-arated by a novel kind of domain wall, a “G-wall”. The idea is implemented by introducing a gaugepotential A µνρ , and its conjugate D , which determines the value of G as an integration constantrather than a fundamental constant. The value of G jumps when one goes through a G -wall. Theprocedure extends one previously developed for the cosmological constant, but the generalizationis far from straightforward: (i) The intrinsic geometry of a G -wall is not the same as seen from itstwo sides, because the second law of black hole thermodynamics mandates that the jump in G mustcause a discontinuity in the scale of length. (ii) The size of the decay step in G is controlled by afunction G ( D ) which may be chosen so as to diminish the value of G towards the asymptote G = 0,without fine tuning. It is shown that: (i) The dynamics of the gravitational field with G treatedas a dynamical variable, coupled to G -walls and matter, follows from an action principle, which isgiven. (ii) A particle that impinges on a G -wall may be refracted or reflected. (iii) The variousforces between two particles change when a G -wall is inserted in between them. (iv) G -walls may benucleated trough tunneling and thermal effects. The semiclassical probabilities are evaluated. (v) Ifthe action principle is constructed properly, the entropy of a black hole increases when the value ofthe gravitational constant is changed through the absorption of a G-wall by the hole. I. INTRODUCTION
A natural scale for the gravitational constant G fromparticle physics would be set by the square of the inversemass of the Higgs boson, 1 /m h ∼ − GeV − . Theobserved value of G is thirty four orders of magnitudesmaller. This enormous difference led Dirac to proposeits “large number hypothesis”[1], where G was consid-ered to change in time, decaying from an initial valueconmensurable with microscopical units, to its value inthe present epoch of the universe. In this paper we pro-pose a mechanism that leads to such a decay. It is basedon assuming that G may take different values within re-gions of the universe separated by a domain wall, whichwe call “ G -wall” for short. The idea is implemented byintroducing a gauge potential A µνρ , and its conjugate D ,which determines the value of G as an integration con-stant rather than a fundamental constant. The value ofthe integration constant jumps when one goes through a G -wall.The procedure extends one previously developed forthe cosmological constant[2–4], but the generalization isfar from straightforward: (i) The intrinsic geometry of a G -wall is not the same as seen from its two sides. Thisis because the second law of black hole thermodynamicsmandates that the jump in G must cause a discontinuityin the scale of length. (ii) The size of the decay step in ∗ [email protected] † andres.gomberoff@uai.cl G is controlled by a function G ( D ) which may be chosenso as to diminish the value of G towards the asymptote G = 0, without fine tuning.The fact that G sets the scale of length is similar tothe way the effective gravitational constant changes inthe Jordan-Brans-Dicke theory[5, 6] or other dilatonictheories; but with the key difference that here the changesin G are driven by the introduction of a 3-form A , andits conjugate D , which do not have any local degrees offreedom.The plan of the paper is the following. Section II is de-voted to introduce the key properties of the G -wall thatseparates regions of space with different values of G . Thebasis for the discussion is the introduction of the conceptof “gravitational units”, that will be key throughout thepaper, and which incorporates the fact that the scale oflength is changed when one crosses the G -wall. Two ef-fects are displayed, namely, (i)the forces (gravitationaland non-gravitational) between two particles change ina distinct manner when a G -wall is inserted in betweenthem,(ii) if a particle impinges on a G -wall, its worldlineis refracted or reflected depending on its velocity. SectionIII deals with the action principle. The action is given,the equations of motion are derived; and the simple casein which the matter is a uniform vacuum energy (cosmo-logical constant of microscopical origin) is dealt with indetail as a preparation for the study of G -wall nucleationin section IV. In that section, the nucleation through tun-neling and thermal activation is discussed and the geom-etry and probability of the corresponding instanton andthermalon are studied. Finally, in section V, it is shownthat the entropy of a black hole increases when it absorbs a r X i v : . [ g r- q c ] A p r a G -wall. It is argued that the second law of black holethermodynamics actually dictates the form of the action. II. ANTICIPATION: IMPRINT OF A G -WALL Before tackling the general action principle, we will an-ticipate in this section, on the basis of a simple Newto-nian argument, two effects which capture the distinctiveimprint of a G -wall. They are: (a) the alteration in theforces (gravitational and non-gravitational) between twoparticles when a G -wall is inserted in between them, and(b) the effect of the wall on the motion of a test particlethat crosses it. A. Effect of a G -wall on the forces betweenparticles. Gravitational units Consider two particles interacting gravitationallythrough the Newtonian potential. The action is givenby I = (cid:90) dt (cid:18) m (cid:126)v + m (cid:126)v + m m Gr (cid:19) , (II.1)where r = | (cid:126)r − (cid:126)r | is the distance between the particles.Next, perform a change of scale of the three fundamen-tal units; time, length and mass, t = G / ˜ t, (cid:126)r = G / ˜ (cid:126)r, m = G − / ˜ m. (II.2)This change sets G equals to unity, keeping the othertwo universal constants (cid:126) and c unaffected, as it leavesthe units of action and speed invariant. We call these“gravitational units”. This same rescaling was discussed,for example, in [7] in the context of the Jordan-Brans-Dicke theory[5, 6].In gravitational units the action (II.1) reads, I = (cid:90) d ˜ t (cid:18) ˜ m (cid:126)v + ˜ m (cid:126)v + ˜ m ˜ m ˜ r (cid:19) . (II.3)The transition from Eq. (II.1) to Eq. (II.3) was achievedassuming that G was constant throughout space andtime. We will now postulate that (II.3) remains validwhen G -walls are present, in which case G is only domain-wise constant, since it changes its value when one crossesa G -wall. That is, we demand that in gravitational unitsthe G -wall becomes invisible in the action .Consider now a G -wall as shown in Fig. 1. Particle 1,of mass m , is fixed at the origin of coordinates, to theleft of the wall, while particle 2, of mass m , is locatedto its right. Let us first evaluate the gravitational forcefelt by particle 2 due to the field produced by particle 1.The potential energy is˜ U = − ˜ m ˜ m ˜ r , zx l l G G G -wall1 2 FIG. 1. Two test particles, labeled 1 and 2, with masses m , m located at different sides of a G -wall, where the Newtonconstant is G , G respectively. The origin of the coordinatesystem is set on particle 1. where ˜ r = ˜ l + ˜ l , is the sum of the tilded lengths ofthe portions, at different sides of the wall, of the lineconnecting particle 1 and particle 2. In terms of theoriginal, untilded “atomic units” the potential energy ofparticle 2 in the field of particle 1 reads U → = G − / ˜ U = − ˜ m ˜ m (cid:16)(cid:113) G G l + l (cid:17) . (II.4)The force felt by the particle 2 is F → = − ∂∂l U → = − m m G / G / (cid:16)(cid:113) G G l + l (cid:17) . (II.5)It is directed along the line joining the particles. Notethat this force does not satisfy the law of action and reac-tion. This was to be expected since the external G -wallbreaks translation and hence momentum conservation.One may perform the same calculation for theCoulomb interaction if the two particles are charged. Onethen has ˜ U = − ˜ e ˜ e ˜ r , but the units of the charge e are those of (action × velocity) / , and therefore˜ e = e, unlike ˜ m = √ Gm . Therefore, in the Coulomb case Eq.(II.4) is replaced by U → = G − / ˜ U = − ˜ e ˜ e (cid:16)(cid:113) G G l + l (cid:17) , (II.6)while (II.5) is replaced by, F → = − ∂∂l U → = − e e (cid:16)(cid:113) G G l + l (cid:17) . (II.7)It is important to realize that, as shown by the aboveCoulomb example, the presence of the G -wall affects allthe interactions between matter on different sides of thewall and not just the gravitational ones. On the otherhand, if one considers experiments realized within a givendomain, only the gravitational interaction will be differ-ent in different domains. For example, the orbital pe-riod of a satellite (in atomic units) in a given elliptic or-bit, specified by its eccentricity (dimensionless) and thelength of its semimajor axis (given in atomic units), willbe proportional to G − / ; whereas for the Coulomb ana-log there will be no G dependence. B. Reflection and refraction of a particle by a G -wall We now extend the notion of gravitational units to arelativistic context and examine what happens to a testparticle when it crosses a G -wall, as it is described inFig. 2. The metric, in gravitational units, ˜ g µν is relatedto g µν by g µν = G ˜ g µν , (II.8)as it follows from the change of coordinates given in Eq.(II.2). We take dimensionless coordinates, so that themetric g µν has dimensions of length squared, while ˜ g µν has units of action.We will demand again that the G -wall be invisible interms of gravitational units. That means that ˜ g µν shouldbe continuous across the wall, which is equivalent to,1 G g (1) µν = 1 G g (2) µν , (II.9)implying that the intrinsic geometry of the G -wall isdifferent as seen from its two sides . This novel featuremakes G -walls essentially different from ordinary domainwalls, across which the intrinsic geometry is continuous.In term of the tilded variables, the action for the par-ticle to propagate from 1 to 2 crossing the G -wall maybe then written as I = − (cid:90) ˜ m (˜ x ) (cid:112) − ˜ g µν d ˜ x µ d ˜ x ν , (II.10)where ˜ m (˜ x ) = (cid:26) m √ G left of the wall, m √ G right of the wall.Thus the problem is equivalent to a particle whosemass changes for m √ G to m √ G when crossing the G -wall. The components of the four-momentum parallel to G G v v FIG. 2. A particle of mass m crosses a G -wall dividing aregion with gravitational constant G from a second regionwith G . In gravitational units, the situation is equivalent tothat of a free particle that changes its mass as crossing theboundary, from ˜ m = mG / to ˜ m = mG / . The speeds inthis diagram are for the case in which G > G . the G -wall are not altered (translation invariance in thetangent plane of the worldsheet of the wall, where theeffect takes place). Conservation of the spatial momen-tum implies that the movement occurs in a plane and theSnell-type law, v sin θ = v sin θ . (II.11)On the other hand, conservation of energy yields,1 − v G = 1 − v G , (II.12)giving the speed of the particle once it emerges to region2. When the particle is moving to the region with smallerNewton constant G > G , Eq. (II.12) indicates that theparticle experiments a boost, v > v . From (II.11) wesee that in that case θ < θ .In the inverse situation, G < G , the particle willdiminish its speed. In that case, Eqs. (II.11) and (II.12)tell us that, for a given v , there is a maximal angle forwhich the particle may cross the wall,sin θ ≤ v (cid:20) − G G (1 − v ) (cid:21) . (II.13)If Eq. (II.13) is not satisfied the particle will experimenta specular bounce off the wall, similar to total internalreflection in geometrical optics. Note, in particular, thatif v is too small, the right hand side of the inequality(II.13) becomes negative, and the particle will bouncefor any angle. Also note that in the critical case, when(II.13) becomes equality, one has θ = π/ III. ACTION PRINCIPLE
We will now include the G -walls as dynamical objects.This will be achieved by generalizing the procedure in-troduced in [2] to promote the cosmological constant toa dynamical variable that changes across a domain wallpossesing a U(1) charge. A. Action
The action takes the form I = I grav + I wall + I matter . (III.14)Here I grav , which depends on the metric field g µν , a threeform potential A µνρ and a pseudoscalar D is given by I grav = 116 π (cid:90) d x (cid:112) − ˜ g ˜ R − (cid:90) ( ∂ α D ) ∗ A α d x. (III.15)The first term in Eq. (III.15) is the Einstein-Hilbert ac-tion for the tilded metric. The second term is built outof the pseudoscalar vector density ∗ A α , ∗ A α = 13! (cid:15) αβγδ A βγδ , dual to the three-form potential A αβγ and the field D .One sees from Eq. (III.15) that D is canonically conju-gate of ∗ A = A .The second term in Eq. (III.14) is an integral over theworldvolume of the G -wall, I wall = (cid:90) A − ˜ µ π (cid:90) (cid:112) − det ˜ γ d σ . (III.16)Here ˜ µ is the tension of the wall in gravitational units,and ˜ γ ab is the induced tilded metric (which is the samefrom both sides of the wall) on it,˜ γ ab = ∂z µ ∂σ a ∂z ν ∂σ b ˜ g µν , (III.17)where the embedding of the wall in spacetime is z µ = z µ ( σ a ), a = 0 , ,
2, with dimensionless parameters σ a .The relation between the metric g µν , which is the onedefining distances in atomic units (sometimes referred toas the metric in the “Jordan frame”), with the tilded met-ric (sometimes referred to as the metric in the “Einsteinframe”) is given by (II.8) but where now G is a functionof D , G = G ( D ), which will be chosen below.The matter action takes the form, I matter = (cid:90) d x (cid:32)L m . (III.18) Here the matter Lagrangian density (cid:32)L m depends on thematter fields and g µν . It becomes a function of D and˜ g µν through g µν = G ( D )˜ g µν . B. Equations of motion
Although we take the physical metric as g µν , it is sim-pler to work with ˜ g µν which will be continuous across thewall, because G -walls are isometrically embedded in ˜ g µν ,and not in g µν . We will, therefore, vary the action taking˜ g µν as an independent field.Varying the action with respect to A we obtain that D is domain-wise constant, jumping across G -walls as D − D = 1 . (III.19)(We have taken in (III.16) the U(1) charge of the wallto be equal to unity). The identification of the two sideswith the subscripts “1” or “2” depends on the orientationof the worldsheets.Varying with respect to D we get ∗ F ≡ ∂ α ∗ A α = − T (cid:112) − ˜ g ddD G ( D ) , (III.20)where T is the trace of the energy momentum tensor ofthe matter (defined in terms of the microscopic metric g µν ).The time derivative of ∗ A i does not appear in Eq.(III.20), which means that the evolution of ∗ A i is arbi-trary, i.e., ∗ A i is pure gauge. As a consequence, ∂ i ∗ A i isalso arbitrary, and therefore the time evolution of ∗ A isarbitrary, and hence it also is pure gauge, with exceptionof the gauge invariant global mode P = (cid:90) ∗ A d x. (III.21)It follows from (III.20) that˙ P = − (cid:90) d xT (cid:112) − ˜ g ddD G ( D ) . Therefore, after the gauge freedom is factored out, thefields
D, A have only one physical degree of freedom(not one per point) of “action-angle” variables, D , P .The variable D , canonically conjugate to P , is domain-wise independent of time (and space).Note that if [ G ] (cid:48)(cid:48) and T are different from zero, onemay use (III.20) to express D in terms of ∗ F and T . Itis then permisible to insert that solution into the action(III.14) to obatin a reduced action that involves A butnot D . This will be the case of interest in the presentwork. However, the non-invertible case is not devoid ofinterest; an analog of G ( D ) = D , whose second deriva-tive vanishes, was used in [3] to interpret the global mode(III.21) as a “cosmic time”, conjugate to the cosmologicalconstant.Lastly, varying with respect to the metric field ˜ g µν gives rise to the Einstein equations for this metric, with˜ G = 1. The energy momentum tensor in them has twocontributions: one from the G -wall, proportional to thetension ˜ µ and supported on its worldsheet; the other, theenergy momentum ˜ T µν of the matter field obtained byvarying I m with respect to ˜ g µν . One recovers the equa-tions in atomic units by rescaling with G ( D ) as in Eq.(II.8). C. Interaction of G -walls with vacuum energy The simplest form of matter one may consider is a con-stant microscopic vacuum energy density u , which givesrise to a cosmological constantΛ = 8 πGu. (III.22)The corresponding matter action is I matter = − (cid:90) d x u √− g = − (cid:90) d x uG (cid:112) − ˜ g. (III.23)By the very definition of atomic units the constant u isof order unity.Now Eq. (III.20) takes the form, ∗ F = u (cid:112) − ˜ g ddD G ( D ) . (III.24)We will choose the function G ( D ) so that: (i) bubblenucleation decreases G , making it vanish asymptoticallywithout ever becoming negative, (ii) no fine tuning isnecessary.Thus, in this view, the present small value of G is stilldecreasing, but at an extremely small rate.Such functions G ( D ) do exist. A simple choice is, G ( D ) = G D , (III.25)and it is the one we will use bellow. Note that G decreases when D increases . (The possibility of such an “attractorpoint” in bubble nucleation has been considered in a dif-ferent setting and through a different mechanism in Ref.[8]) IV. PRODUCTION OF G -WALLS BYTUNNELING AND THERMAL ACTIVATION.INSTANTON AND THERMALON In [2] and in [4], the cosmological constant was re-laxed by nucleating membranes due to quantum tunnel-ing (“going through the potential barrier”) and thermalactivation (“jumping over the potential barrier”) respec-tively. The same phenomena occurs here, where nucle-ation of G -walls relaxes the gravitational constant.We will now study the probability for nucleating G -walls through both processes in the semiclassical approx-imation. In doing so we will consider only the uniform vacuum energy density u case discussed above, but wewill keep the function G ( D ) generic.The probability is of the form, P = C exp( I E ) , (IV.26)where I E is the Euclidean action evaluated on an ap-propriate extremum, and C is a slowly varying func-tion. The Euclidean action is obtained from the ex-ponent iI of the exponential in the Lorentzian pathintegral, by replacing in it ( τ, x , x i , A ij , A ijk , D ) by( − iτ, − ix , x i , − A ij , iA ijk , D ), and demanding the newvariables to be real. This gives I E = 116 π (cid:90) d x (cid:112) ˜ g ˜ R + (cid:90) d x ( ∂ α D ) ∗ A α − (cid:90) A − ˜ µ π (cid:90) d σ (cid:112) det ˜ γ − (cid:90) d x uG (cid:112) − ˜ g , (IV.27)which is understood to be a functional of ˜ g µν , A µνρ , D ,and z µ . Note that the action is linear in A , and thereforewhen evaluating it on-shell, the second and third termcancel. The only D dependence left is through G ( D ) inthe matter term of (IV.27). The probability of nucleating G -walls will, therefore, be expressible in terms of G andits change ∆ G due to the nucleation. The change ∆ G ina nucleation will be determined through the function of G ( D ) by the U (1) charge of the bubble, taken here byconvention equal to unity, ∆ D = 1, (Eq. (III.19)). A. Euclidean worldsheet of a spherical G -wall One expects the extremum relevant for the semiclassi-cal approximation to have the highest possible symmetry.Since we will be interested in including black holes, wewill take in the succeeding discussion the symmetry tobe just ordinary spherical symmetry SO(3), rather thanSO(4). The symmetry will become SO(4) when there isno black hole.Consider a spherical G -wall, dividing spacetime in tworegions, the “interior” (“-”) and the “exterior” (“+”),such that the metric on each side is of Schwarzschild-deSitter form, d ˜ s ± = ˜ f ± d ˜ t ± + ˜ f − ± d ˜ r + ˜ r d Ω , (IV.28)˜ f ± = 1 − M ± ˜ r − ˜ r ˜ l ± . (IV.29)(In order to make direct contact with previous results[4], we use in this subsection time and radial coordinateswith dimensions of length).In (IV.29),1˜ l ± = ˜Λ ± π u ± = 8 π G ± u, (IV.30)and the coordinate ˜ t ± is defined such that increases an-ticlockwise around the cosmological horizon. The wall isparameterized by,˜ r = R ( τ ) , ˜ t ± = T ± ( τ ) , (IV.31)with τ being the proper time,˜ f ± ˙ T ± + ˜ f − ± ˙ R = 1 . (IV.32)The matching in the geometries at the joining mem-brane gives a first integral for the equation of motion of R ( τ ), (cid:113) ˜ f − − ˙ R − (cid:113) ˜ f − ˙ R = σ ˜ µR. (IV.33)Here σ = 1, when the cosmological horizon is in the ex-terior, which will be the case for the instanton discussedbelow; and σ = − B. Radius
There are two types of solutions of Eq.(IV.33) that areof interest here. The instanton described in [2] for the nu-cleation of membranes by tunneling; and the thermalon,a static solution found in [4] for the nucleation by ther-mal activation. In both cases it is necessary to determinethe radius of formation R = ρ which occurs when ˙ R = 0.From (IV.33), at that instant,∆ ˜ M ≡ ˜ M − − ˜ M + = 12 ( α − ˜ µ ) ρ − σ ˜ µ ˜ f + ρ , (IV.34)where α = 83 πu ( G − G − ) . (IV.35)For the instanton σ = 1 in Eq. (IV.33). Here there isno black hole, hence ˜ M ± = 0, and the geometry is thatof de Sitter on each side of the G -wall. In this case Eq.(IV.34) is quadratic, and it has a solution if and only if α > ˜ µ >
0, that is, when G > G − , (IV.36)so that the notation in (IV.35) is justified. The radius offormation is given by ρ = 2˜ µ (cid:32) [ α − ˜ µ ] + 4˜ µ ˜ l (cid:33) − / . (IV.37)The function R ( τ ) is a “bounce” going from R = 0 toa maximum R = ρ and back to R = 0. The Euclidean G -wall worldsheet is a three-sphere of radius ρ where thetwo de Sitter geometries are glued together. Once the wall materializes its radius increases without limit (see[2] for details).For the thermalon ∆ ˜ M does not vanish. Its valueis determined by demanding the right hand side of Eq.(IV.34) to be an extremum in ρ , which is equivalent torequiring ¨ R = 0. Once this value is inserted on the lefthand side of (IV.34) the resulting equation cannot besolved analytically, but can be analyzed graphically. Onemay show that solutions with positive α also exist. Thatanalysis is given in [4] and will be not repeated here.The thermalon is static, therefore ˙ R = 0, and the G -wall stays always at R = ρ unless it is perturbed.Of particular interest is the “cosmological thermalon” inthermal equilibrium with the cosmological horizon. Forthat solution the cosmological horizon is in the interior sothat σ = − G -wall will collapseto a black hole, even if originally there is none ( M + = 0, M − (cid:54) = 0). C. Probability
It is simpler to evaluate the on-shell action in theHamiltonian formalism. This is so because: (i) withineach of the domains separated by the wall one may usea time coordinate in which the metric is static, thereforethe π ij ˙ g ij term vanishes. (ii) The total “bulk” Hamil-tonian, which is a linear combination of the constraintsvanishes, because on-shell the constraints hold.(iii) Whena horizon is present in the interior (final region) one justadds one fourth of its area, ˜ A − in the tilded metric[9].Therefore the on-shell action, reduces to the p ˙ q term ofthe G -wall, which, rewritten in term of the velocities isgiven by, I E = 14 ˜ A − − (cid:90) A − ˜ µ π (cid:90) d σ (cid:112) det ˜ γ. (IV.38)The second term in this expression is ill defined, be-cause the value of A jumps as one crosses the G -wall.The regularized value of the integral must be taken tobe, (cid:90) A = ∗ F on V (4) − , (IV.39)where V (4) − is the volume of the interior region and ∗ F on is the value of ∗ F on the membrane, which may be de-fined by thickening the membrane and using the meanvalue theorem as ∗ F on = (cid:18) D + − D − (cid:19) (cid:90) D + D − ∗ F dD. (IV.40)Eq. (IV.39) may be justified by realizing that the varia-tion of its right hand side with respect to the membranecoordinates should give the “electric force” q ∗ F on , where q = D + − D − = 1. On the other hand, the Euclideanvalue of ∗ F is given by the counterpart of (IV.41), ∗ F = − u (cid:112) ˜ g ddD G ( D ) . (IV.41)Inserting this in (IV.40), and carrying out the integra-tion, yields (cid:90) A = uV (4) − ( G − − G ) . (IV.42)The final regularized action is then I E = 14 ( ˜ A − − ˜ A + ) − ˜ µ π V (3) − uV (4) − ( G − − G ) , (IV.43)where V (3) is the three-volume of the history of the G -wall, and ˜ A + is the area of the horizon that the interiorregion would have if the G -wall were absent. As explainedin Sec. IV of [4], the substractions of G and of ˜ A + / G -walls with µ = u = 0 isequal to unity.Eq. (IV.43) is the same that was obtained in Ref.[4] in the context of the cosmological constant problem.Therefore all the analysis carried out there for the prob-ability in various limiting cases can be taken over simplyby substituting Eq. (IV.35) for α in that reference.Instead we would like to devote some space to bring outhow the formula for the probability captures in a nutshellthe fact that, in the case of G, the decay process relatestwo different scales, a feature which has no analog in thecosmological constant case. This may be seen in the sim-plest manner for the instanton in the approximation inwhich the nucleation radius ρ is much smaller that thedS radius ˜ l + . In that case we may approximate the in-terior volume by its flat space expression, V (4) − = π ρ ,whereas V = 2 π ρ . Therefore the action takes the form I instantonE = − ˜ µπ ρ + 3 π α ρ . (IV.44)Minimizing this action with respect to ρ we obtain theradius of formation, ρ = 2˜ µα , which could also be obtained directly from Eq. (IV.37).Inserting this in (IV.44) gives, I instantonE = − π ˜ µ α . (IV.45)The probability of formation is therefore given by, P ∼ exp (cid:18) (cid:126) I instantonE (cid:19) == exp (cid:34) − (cid:18) (cid:19) ˜ µ π (cid:126) u ( G − G − ) (cid:35) , (IV.46) where we have substituted for α in (IV.45) its value(IV.35). If D >>
1, which begins to be true after a fewsteps, and becomes more and more valid throughout thepresent time, then G + ≈ G − = G , and we may writethis, P ∼ exp (cid:18) − (cid:20) π | (log G ) (cid:48) | (cid:21) (cid:20) ˜ µ (cid:126) u G (cid:21)(cid:19) . (IV.47)The first factor in the exponent is of order unity, whereasthe second is more illuminating. By using ˜ u = G u ,˜ µ = G / µ , it maybe rewritten as, µ (cid:126) u = ˜ µ (cid:126) ˜ u . (IV.48)Each of the two sides of the equality contains a termwhich is natural (i.e., of order unity) in atomic units andanother which is natural in gravitational units. For ex-ample, on the left hand side, µ is of gravitational naturebecause it is the tension of a G -wall, so its tilded valueis always of order unity but its untilded value dependson the epoch: it is of order unity at the beginning, butvery large at present. On the other hand u , which is ofatomic nature is always of order unity. Thus, the proba-bility (IV.47) is of order unity at the beginning and verysmall at present. The same conclusion is of course ob-tained looking at the right hand side. Now ˜ µ is alwaysof order unity, but ˜ u depends on the epoch. V. THE GRAVITATIONAL CONSTANT AS ATHERMODYNAMIC BLACK HOLEPARAMETER
When a black hole metric is expressed in terms ofthe mass, angular momentum, and other charges, it de-pends explicitly, in addition, on the gravitational con-stant G and the cosmological constant Λ. The standardblack hole thermodynamics theorems (see, for example,[10]) assume that both G and Λ are universal constants;therefore the question of whether the black hole entropycan only increase in a irreversible process has to be re-analyzed when G and Λ are allowed to vary. For thecosmological constant, this analysis was performed in thesimple case of spherical symmetry in Ref. [11] and thequestion was answered in the affirmative. It is the pur-pose of this section to address the question for G .The issue at hand becomes already manifest for thecase of a Schwarzschild black hole, ds = − (cid:18) − M Gr (cid:19) dt + (cid:18) − M Gr (cid:19) − dr + r d Ω , (V.49)for which the entropy is given by S = 4 πGM (cid:126) . (V.50)When the hole absorbs a G -wall both G and M willchange. It may happen that the value of G diminishesas it indeed is the case, for example, for the thermalondiscussed in subsection IV.B (recall Eq. (IV.36)). Now,if we had coupled the G -wall to g µν as a standard do-main wall, the mass M of the hole would increase afterabsorption of the wall, since the latter would have pos-itive energy; but there would be no guarantee that theincreasing M would be sufficient to overcompensate thedecreasing G so as to make the entropy increase, sinceboth changes are independent. Thus, the second law ofblack hole thermodynamics could be easily violated andone would face a major difficulty.The way out is provided by realizing that in what wehave called gravitational units the entropy (V.50) reads, S = 4 π ˜ M (cid:126) , (V.51)and therefore if one couples the domain wall in the stan-dard manner in gravitational units , as in (III.14), theproblem is solved because now ˜ M increases and so doesthe entropy. The argument holds as well for a generalblack hole, because in gravitational units, the standardtheorems apply.The value of D , which determines G , may be thoughtof as a charge, which counts - up to an additive constant- the number of G -walls absorbed by the hole. Its conju-gate chemical potential is the integral of A θφ at infinity(or at the cosmological horizon), when one demands A θφ to vanish at the black hole horizon.It is quite remarkable that the second law of blackhole thermodynamics should dictate how a dynamical G must be incorporated in the action principle if one wantsto relate the cosmological and microscopical scales. Butperhaps it should not be surprising, since after all S = c (cid:126) G (area of black hole horizon) , is the one formula where those two scales appear explic-itly rather than just being related through dimensionalanalysis. ACKNOWLEDGMENTS
The Centro de Estudios Cient´ıficos (CECs) is fundedby the Chilean Government through the Centers of Ex-cellence Base Financing Program of Conicyt. C.B. wishesto thank the Alexander von Humboldt Foundation for aHumboldt Research Award. The work of A.G. was par-tially supported by Fondecyt (Chile) Grant [1] P. A. M. Dirac, Nature , 323 (1937).[2] J. D. Brown and C. Teitelboim, Nucl.Phys.
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B158 , 293 (1985). For a standard domain wall coupling, the Hilbert term in the ac-tion (III.14), √− ˜ g ˜ R , would be replaced by G ( D ) − √− gR . No˜ g µν would be introduced, the factor G ( D ) − would remain “out-side of the curvature”. Straightforward analogs of the action soobtained may be written to make dynamical, for example, thecosmological constant, any parameter appearing in a field theoryLagrangian (coupling constants, masses) or the string tension.In none of these cases there is a problem such as the one that arises with a dynamical G in connection with black hole thermo-dynamics.For the case of the cosmological constant, if one setsΛ( D ) = 4 πGe D + λ in the Hilbert Lagrangian, one obtains theaction employed in [2]. (Here e is the charge of the standard do-main wall and λλ